The supercritical Galton-Watson process in varying environments

AbstractLet {Zn} be a supercritical Galton-Watson process in varying environments. It is known that Zn when normed by its mean EZn converges almost surely to a finite random variable W. It is possible, however, for such a process to exhibit more than one rate of growth so that in particular {W > 0} need not coincide with {Zn → ∞}. Here a natural sufficient condition is given which ensures that this cannot happen. Under a weaker condition it is shown that the possible rates of growth cannot differ very much in that {ZnEZn}1n → 1 on {Zn → ∞}